Theorem lspsolv 21304

Description: If 𝑋 is in the span of 𝐴 ∪ {𝑌} but not 𝐴, then 𝑌 is in the span of 𝐴 ∪ {𝑋}. (Contributed by Mario Carneiro, 25-Jun-2014.)

Hypotheses

Ref	Expression
lspsolv.v	⊢ 𝑉 = (Base‘𝑊)
lspsolv.s	⊢ 𝑆 = (LSubSp‘𝑊)
lspsolv.n	⊢ 𝑁 = (LSpan‘𝑊)

Assertion

Ref	Expression
lspsolv	⊢ ((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) → 𝑌 ∈ (𝑁‘(𝐴 ∪ {𝑋})))

Proof of Theorem lspsolv

Dummy variables 𝑟 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lspsolv.v	. . 3 ⊢ 𝑉 = (Base‘𝑊)
2		lspsolv.s	. . 3 ⊢ 𝑆 = (LSubSp‘𝑊)
3		lspsolv.n	. . 3 ⊢ 𝑁 = (LSpan‘𝑊)
4		eqid 2766	. . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
5		eqid 2766	. . 3 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6		eqid 2766	. . 3 ⊢ (+g‘𝑊) = (+g‘𝑊)
7		eqid 2766	. . 3 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
8		eqid 2766	. . 3 ⊢ {𝑧 ∈ 𝑉 ∣ ∃𝑟 ∈ (Base‘(Scalar‘𝑊))(𝑧(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴)} = {𝑧 ∈ 𝑉 ∣ ∃𝑟 ∈ (Base‘(Scalar‘𝑊))(𝑧(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴)}
9		lveclmod 21264	. . . 4 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
10	9	adantr 486	. . 3 ⊢ ((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) → 𝑊 ∈ LMod)
11		simpr1 1213	. . 3 ⊢ ((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) → 𝐴 ⊆ 𝑉)
12		simpr2 1214	. . 3 ⊢ ((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) → 𝑌 ∈ 𝑉)
13		simpr3 1215	. . . 4 ⊢ ((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) → 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))
14	13	eldifad 3920	. . 3 ⊢ ((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) → 𝑋 ∈ (𝑁‘(𝐴 ∪ {𝑌})))
15	1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 14	lspsolvlem 21303	. 2 ⊢ ((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) → ∃𝑟 ∈ (Base‘(Scalar‘𝑊))(𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))
16	4	lvecdrng 21263	. . . . . . 7 ⊢ (𝑊 ∈ LVec → (Scalar‘𝑊) ∈ DivRing)
17	16	ad2antrr 739	. . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (Scalar‘𝑊) ∈ DivRing)
18		simprl 783	. . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝑟 ∈ (Base‘(Scalar‘𝑊)))
19	10	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝑊 ∈ LMod)
20	12	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝑌 ∈ 𝑉)
21		eqid 2766	. . . . . . . . . . . . 13 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
22		eqid 2766	. . . . . . . . . . . . 13 ⊢ (0g‘𝑊) = (0g‘𝑊)
23	1, 4, 7, 21, 22	lmod0vs 21053	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → ((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌) = (0g‘𝑊))
24	19, 20, 23	syl2anc 596	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → ((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌) = (0g‘𝑊))
25	24	oveq2d 7439	. . . . . . . . . 10 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑋(+g‘𝑊)((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌)) = (𝑋(+g‘𝑊)(0g‘𝑊)))
26	11	adantr 486	. . . . . . . . . . . . . . 15 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝐴 ⊆ 𝑉)
27	20	snssd 4757	. . . . . . . . . . . . . . 15 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → {𝑌} ⊆ 𝑉)
28	26, 27	unssd 4148	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝐴 ∪ {𝑌}) ⊆ 𝑉)
29	1, 3	lspssv 21141	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ LMod ∧ (𝐴 ∪ {𝑌}) ⊆ 𝑉) → (𝑁‘(𝐴 ∪ {𝑌})) ⊆ 𝑉)
30	19, 28, 29	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑁‘(𝐴 ∪ {𝑌})) ⊆ 𝑉)
31	30	ssdifssd 4104	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)) ⊆ 𝑉)
32	13	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))
33	31, 32	sseldd 3941	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝑋 ∈ 𝑉)
34	1, 6, 22	lmod0vrid 21051	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑋(+g‘𝑊)(0g‘𝑊)) = 𝑋)
35	19, 33, 34	syl2anc 596	. . . . . . . . . 10 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑋(+g‘𝑊)(0g‘𝑊)) = 𝑋)
36	25, 35	eqtrd 2801	. . . . . . . . 9 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑋(+g‘𝑊)((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌)) = 𝑋)
37	36, 32	eqeltrd 2866	. . . . . . . 8 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑋(+g‘𝑊)((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌)) ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))
38	37	eldifbd 3921	. . . . . . 7 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → ¬ (𝑋(+g‘𝑊)((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))
39		simprr 785	. . . . . . . . 9 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))
40		oveq1 7430	. . . . . . . . . . 11 ⊢ (𝑟 = (0g‘(Scalar‘𝑊)) → (𝑟( ·𝑠 ‘𝑊)𝑌) = ((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌))
41	40	oveq2d 7439	. . . . . . . . . 10 ⊢ (𝑟 = (0g‘(Scalar‘𝑊)) → (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) = (𝑋(+g‘𝑊)((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌)))
42	41	eleq1d 2851	. . . . . . . . 9 ⊢ (𝑟 = (0g‘(Scalar‘𝑊)) → ((𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴) ↔ (𝑋(+g‘𝑊)((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴)))
43	39, 42	syl5ibcom 248	. . . . . . . 8 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑟 = (0g‘(Scalar‘𝑊)) → (𝑋(+g‘𝑊)((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴)))
44	43	necon3bd 2975	. . . . . . 7 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (¬ (𝑋(+g‘𝑊)((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴) → 𝑟 ≠ (0g‘(Scalar‘𝑊))))
45	38, 44	mpd 16	. . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝑟 ≠ (0g‘(Scalar‘𝑊)))
46		eqid 2766	. . . . . . 7 ⊢ (.r‘(Scalar‘𝑊)) = (.r‘(Scalar‘𝑊))
47		eqid 2766	. . . . . . 7 ⊢ (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑊))
48		eqid 2766	. . . . . . 7 ⊢ (invr‘(Scalar‘𝑊)) = (invr‘(Scalar‘𝑊))
49	5, 21, 46, 47, 48	drnginvrl 20897	. . . . . 6 ⊢ (((Scalar‘𝑊) ∈ DivRing ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑟 ≠ (0g‘(Scalar‘𝑊))) → (((invr‘(Scalar‘𝑊))‘𝑟)(.r‘(Scalar‘𝑊))𝑟) = (1r‘(Scalar‘𝑊)))
50	17, 18, 45, 49	syl3anc 1398	. . . . 5 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (((invr‘(Scalar‘𝑊))‘𝑟)(.r‘(Scalar‘𝑊))𝑟) = (1r‘(Scalar‘𝑊)))
51	50	oveq1d 7438	. . . 4 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → ((((invr‘(Scalar‘𝑊))‘𝑟)(.r‘(Scalar‘𝑊))𝑟)( ·𝑠 ‘𝑊)𝑌) = ((1r‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌))
52	5, 21, 48	drnginvrcl 20894	. . . . . 6 ⊢ (((Scalar‘𝑊) ∈ DivRing ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑟 ≠ (0g‘(Scalar‘𝑊))) → ((invr‘(Scalar‘𝑊))‘𝑟) ∈ (Base‘(Scalar‘𝑊)))
53	17, 18, 45, 52	syl3anc 1398	. . . . 5 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → ((invr‘(Scalar‘𝑊))‘𝑟) ∈ (Base‘(Scalar‘𝑊)))
54	1, 4, 7, 5, 46	lmodvsass 21045	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ (((invr‘(Scalar‘𝑊))‘𝑟) ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑌 ∈ 𝑉)) → ((((invr‘(Scalar‘𝑊))‘𝑟)(.r‘(Scalar‘𝑊))𝑟)( ·𝑠 ‘𝑊)𝑌) = (((invr‘(Scalar‘𝑊))‘𝑟)( ·𝑠 ‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)))
55	19, 53, 18, 20, 54	syl13anc 1399	. . . 4 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → ((((invr‘(Scalar‘𝑊))‘𝑟)(.r‘(Scalar‘𝑊))𝑟)( ·𝑠 ‘𝑊)𝑌) = (((invr‘(Scalar‘𝑊))‘𝑟)( ·𝑠 ‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)))
56	1, 4, 7, 47	lmodvs1 21048	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → ((1r‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌) = 𝑌)
57	19, 20, 56	syl2anc 596	. . . 4 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → ((1r‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑌) = 𝑌)
58	51, 55, 57	3eqtr3d 2809	. . 3 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (((invr‘(Scalar‘𝑊))‘𝑟)( ·𝑠 ‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) = 𝑌)
59	33	snssd 4757	. . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → {𝑋} ⊆ 𝑉)
60	26, 59	unssd 4148	. . . . 5 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝐴 ∪ {𝑋}) ⊆ 𝑉)
61	1, 2, 3	lspcl 21134	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ (𝐴 ∪ {𝑋}) ⊆ 𝑉) → (𝑁‘(𝐴 ∪ {𝑋})) ∈ 𝑆)
62	19, 60, 61	syl2anc 596	. . . 4 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑁‘(𝐴 ∪ {𝑋})) ∈ 𝑆)
63	1, 4, 7, 5	lmodvscl 21036	. . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑌 ∈ 𝑉) → (𝑟( ·𝑠 ‘𝑊)𝑌) ∈ 𝑉)
64	19, 18, 20, 63	syl3anc 1398	. . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑟( ·𝑠 ‘𝑊)𝑌) ∈ 𝑉)
65		eqid 2766	. . . . . . 7 ⊢ (-g‘𝑊) = (-g‘𝑊)
66	1, 6, 65	lmodvpncan 21073	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ (𝑟( ·𝑠 ‘𝑊)𝑌) ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → (((𝑟( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)𝑋)(-g‘𝑊)𝑋) = (𝑟( ·𝑠 ‘𝑊)𝑌))
67	19, 64, 33, 66	syl3anc 1398	. . . . 5 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (((𝑟( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)𝑋)(-g‘𝑊)𝑋) = (𝑟( ·𝑠 ‘𝑊)𝑌))
68	1, 6	lmodcom 21066	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ (𝑟( ·𝑠 ‘𝑊)𝑌) ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → ((𝑟( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)𝑋) = (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)))
69	19, 64, 33, 68	syl3anc 1398	. . . . . . 7 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → ((𝑟( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)𝑋) = (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)))
70		ssun1 4134	. . . . . . . . . 10 ⊢ 𝐴 ⊆ (𝐴 ∪ {𝑋})
71	70	a1i 11	. . . . . . . . 9 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝐴 ⊆ (𝐴 ∪ {𝑋}))
72	1, 3	lspss 21142	. . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ (𝐴 ∪ {𝑋}) ⊆ 𝑉 ∧ 𝐴 ⊆ (𝐴 ∪ {𝑋})) → (𝑁‘𝐴) ⊆ (𝑁‘(𝐴 ∪ {𝑋})))
73	19, 60, 71, 72	syl3anc 1398	. . . . . . . 8 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑁‘𝐴) ⊆ (𝑁‘(𝐴 ∪ {𝑋})))
74	73, 39	sseldd 3941	. . . . . . 7 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘(𝐴 ∪ {𝑋})))
75	69, 74	eqeltrd 2866	. . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → ((𝑟( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)𝑋) ∈ (𝑁‘(𝐴 ∪ {𝑋})))
76	1, 3	lspssid 21143	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ (𝐴 ∪ {𝑋}) ⊆ 𝑉) → (𝐴 ∪ {𝑋}) ⊆ (𝑁‘(𝐴 ∪ {𝑋})))
77	19, 60, 76	syl2anc 596	. . . . . . 7 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝐴 ∪ {𝑋}) ⊆ (𝑁‘(𝐴 ∪ {𝑋})))
78		snidg 4631	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ {𝑋})
79		elun2 4139	. . . . . . . 8 ⊢ (𝑋 ∈ {𝑋} → 𝑋 ∈ (𝐴 ∪ {𝑋}))
80	33, 78, 79	3syl 19	. . . . . . 7 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝑋 ∈ (𝐴 ∪ {𝑋}))
81	77, 80	sseldd 3941	. . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝑋 ∈ (𝑁‘(𝐴 ∪ {𝑋})))
82	65, 2	lssvsubcl 21102	. . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ (𝑁‘(𝐴 ∪ {𝑋})) ∈ 𝑆) ∧ (((𝑟( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)𝑋) ∈ (𝑁‘(𝐴 ∪ {𝑋})) ∧ 𝑋 ∈ (𝑁‘(𝐴 ∪ {𝑋})))) → (((𝑟( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)𝑋)(-g‘𝑊)𝑋) ∈ (𝑁‘(𝐴 ∪ {𝑋})))
83	19, 62, 75, 81, 82	syl22anc 852	. . . . 5 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (((𝑟( ·𝑠 ‘𝑊)𝑌)(+g‘𝑊)𝑋)(-g‘𝑊)𝑋) ∈ (𝑁‘(𝐴 ∪ {𝑋})))
84	67, 83	eqeltrrd 2867	. . . 4 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (𝑟( ·𝑠 ‘𝑊)𝑌) ∈ (𝑁‘(𝐴 ∪ {𝑋})))
85	4, 7, 5, 2	lssvscl 21113	. . . 4 ⊢ (((𝑊 ∈ LMod ∧ (𝑁‘(𝐴 ∪ {𝑋})) ∈ 𝑆) ∧ (((invr‘(Scalar‘𝑊))‘𝑟) ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑟( ·𝑠 ‘𝑊)𝑌) ∈ (𝑁‘(𝐴 ∪ {𝑋})))) → (((invr‘(Scalar‘𝑊))‘𝑟)( ·𝑠 ‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘(𝐴 ∪ {𝑋})))
86	19, 62, 53, 84, 85	syl22anc 852	. . 3 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → (((invr‘(Scalar‘𝑊))‘𝑟)( ·𝑠 ‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘(𝐴 ∪ {𝑋})))
87	58, 86	eqeltrrd 2867	. 2 ⊢ (((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑋(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑌)) ∈ (𝑁‘𝐴))) → 𝑌 ∈ (𝑁‘(𝐴 ∪ {𝑋})))
88	15, 87	rexlimddv 3175	1 ⊢ ((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) → 𝑌 ∈ (𝑁‘(𝐴 ∪ {𝑋})))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ≠ wne 2961  ∃wrex 3092  {crab 3419   ∖ cdif 3905   ∪ cun 3906   ⊆ wss 3908  {csn 4594  ‘cfv 6543  (class class class)co 7423  Basecbs 17294  +_gcplusg 17335  ._rcmulr 17336  Scalarcsca 17338   ·_𝑠 cvsca 17339  0_gc0g 17517  -_gcsg 19033  1_rcur 20294  inv_rcinvr 20502  DivRingcdr 20864  LModclmod 21018  LSubSpclss 21089  LSpanclspn 21129  LVecclvec 21260

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745  ax-cnex 11174  ax-resscn 11175  ax-1cn 11176  ax-icn 11177  ax-addcl 11178  ax-addrcl 11179  ax-mulcl 11180  ax-mulrcl 11181  ax-mulcom 11182  ax-addass 11183  ax-mulass 11184  ax-distr 11185  ax-i2m1 11186  ax-1ne0 11187  ax-1rid 11188  ax-rnegex 11189  ax-rrecex 11190  ax-cnre 11191  ax-pre-lttri 11192  ax-pre-lttrn 11193  ax-pre-ltadd 11194  ax-pre-mulgt0 11195

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-nel 3068  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-int 4918  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7872  df-1st 7995  df-2nd 7996  df-tpos 8231  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-er 8703  df-en 8953  df-dom 8954  df-sdom 8955  df-pnf 11263  df-mnf 11264  df-xr 11265  df-ltxr 11266  df-le 11267  df-sub 11461  df-neg 11462  df-nn 12252  df-2 12321  df-3 12322  df-sets 17249  df-slot 17267  df-ndx 17279  df-base 17295  df-ress 17316  df-plusg 17348  df-mulr 17349  df-0g 17519  df-mgm 18723  df-sgrp 18806  df-mnd 18822  df-grp 19034  df-minusg 19035  df-sbg 19036  df-cmn 19883  df-abl 19884  df-mgp 20248  df-rng 20262  df-ur 20295  df-ring 20348  df-oppr 20452  df-dvdsr 20472  df-unit 20473  df-invr 20503  df-drng 20866  df-lmod 21020  df-lss 21090  df-lsp 21130  df-lvec 21261

This theorem is used by:  lssacsex  21305  lspsnat  21306  lsppratlem1  21308  lsppratlem3  21310  lsppratlem4  21311  lbsextlem4  21322  lindsadd  38305  lindsenlbs  38307